Grothendieck-Serre conjecture for adjoint groups of types E_6 and E_7 and for certain classical groups

TL;DR

This paper proves the Grothendieck-Serre conjecture for certain algebraic groups over semi-local rings, confirming that torsors trivial over the fraction field are already trivial over the ring itself.

Contribution

It extends the validity of the Grothendieck-Serre conjecture to groups of types E_6, E_7, and certain classical groups over specified rings.

Findings

Kernel of H^1_{et}(R,H) to H^1_{et}(K,H) map is trivial for the specified groups.

Confirms conjecture for new classes of algebraic groups over semi-local rings.

Builds on recent series of papers by the authors and Vavilov.

Abstract

Assume that R is a semi-local regular ring containing an infinite perfect field, or that R is a semi-local ring of several points on a smooth scheme over an infinite field. Let K be the field of fractions of R. Let H be a strongly inner adjoint simple algebraic group of type E_6 or E_7 over R, or any twisted form of one of the split groups of classical type O^+_{n,R}, n>=4; PGO_{n,R}, n>=4; PSp_{2n,R}, n>=2; PGL_{n,R}, n>=2. We prove that the kernel of the map H^1_{et}(R,H)-> H^1_{et}(K,H) induced by the inclusion of R into K is trivial. This continues the recent series of papers by the authors and N. Vavilov on the Grothendieck--Serre conjecture.